The purpose of this Warm-up is to help students visualize circumference as a linear measurement, in preparation for examining the relationship between diameter and circumference in the next activity. Some students may be able to imagine unrolling the tube into a rectangle in order to compare its length and width. Other students may benefit from hands-on experience with a real toilet paper tube.
Launch
Arrange students in groups of 2. Display the image for all to see. Ask students to indicate when they have reasoning to support their response. Give students 1 minute of quiet think time and then time to share their thinking with their group.
Activity
None
Clare wonders if the height of the toilet paper tube or the distance around the tube is greater. What information would she need in order to solve the problem? How could she find this out?
Three different views of a circular shaped toilet paper tube. The first view is of the vertical height of the tube. The second view is of the circular base of the tube. The third view is of both the base and the height of the tube.
Student Response
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Building on Student Thinking
Students may not understand what is meant by the height of the tube because it can sit in two different ways. Point these students to the first picture of the tube, and ask them to identify the height as the tube is sitting in that picture.
Activity Synthesis
The goal of this discussion is for students to internalize that the distance around a circle is a length, just like the height of an object is. As a one-dimensional measurement, it can be measured using a ruler.
Survey students on which length they think is greater. Consider displaying the image again for reference while students are explaining their reasoning. To involve more students in the conversation, consider asking questions like:
"What was important to you when making your decision?"
"Did anyone think about the measurements in a different way?"
"Do you agree or disagree? Why?"
"What information do you think would help you make a better decision? If an actual toilet paper tube is available, demonstrate measuring the tube by unrolling the tube to measure the circumference and height."
If an actual toilet paper tube is available, consider demonstrating unrolling the tube to measure the circumference in a straight line and compare it to the height.
12.2
Activity
Standards Alignment
Building On
6.SP.B.5.c
Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
In this activity, students measure the diameter and circumference of different circular objects and graph their measurements. This echoes the structure of a previous activity in which they measured squares. Students notice that the two quantities appear to be proportional to each other. Based on the graph, they can estimate that the constant of proportionality is close to 3. From the table they may be able to estimate that it is a little bigger than 3.
This activity provides evidence that there is a constant of proportionality between the circumference of a circle and its diameter. The best precision we can expect for the constant of proportionality in this activity is “around 3” or possibly “a little bit bigger than 3.” As students measure multiple circles and notice patterns in their measurements, they express regularity in repeated reasoning (MP8).
Monitor for students who recognize that the relationship between diameter and circumference appears to be proportional, including:
Notice that the points on the graph appear to lie on a line through the origin.
Use the graph to estimate a constant of proportionality.
Divide the values in the table to estimate a constant of proportionality.
Plan to have students present in this order to support moving from less precise to more precise estimation.
In the digital version of the activity, students use applets to measure the circumference of various circles and plot their measurements on the coordinate plane. The first applet allows students to see an animation of the circumference being unwrapped along a ruler. The second applet allows students to enter values into a table and see the points plotted on a coordinate plane. Use the digital version if physical circular objects are not available.
Activity Synthesis
The purpose of this discussion is for students to see that circumference is proportional to diameter and that the constant of proportionality is close to 3.
Invite previously selected students to share their observations. Sequence the discussion of the observations in the order listed in the activity narrative. If possible, record and display their work for all to see. Connect the different responses to the learning goals by asking questions such as:
“What do the representations have in common? How are they different?”
“What do the representations tell us about the relationship between the diameter and circumference of different circles?”
“How does the proportional relationship show up in each representation?”
The key takeaway is that the relationship between diameter and circumference appears to be proportional.
If no student mentions it, invite students to estimate the constant of proportionality by dividing the values on each row of the table. For example,
object
diameter (cm)
circumference (cm)
water bottle
7.2
23.3
3.236
bowl
22.8
71.5
3.136
DVD
12.0
37.0
3.083
medicine bottle
3.1
10.5
3.387
Ask students why these numbers might not be exactly the same (measurement error, rounding). Use the average of the quotients, rounded to one or two decimal places, to come up with a “working value” for the constant of proportionality, such as 3.1. This class-generated constant of proportionality will be used in the next activity. There’s no need to mention pi or its usual approximations yet.
If time permits, consider discussing the accuracy of measurements for circumference and diameter. Measuring the diameter to the nearest tenth of a centimeter can be done pretty reliably with a ruler. Measuring the circumference of a circle to the nearest tenth of a centimeter may or may not be reliable, depending on the method used. Wrapping a flexible measuring tape around the object is likely the most accurate method for measuring the circumference of a circle.
12.3
Activity
Standards Alignment
Building On
Addressing
7.G.B.4
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
In this activity, students apply the proportional relationship between circumference and diameter to calculate unknown measurements of circles. They use the constant of proportionality they estimated in the previous activity. Then, the teacher introduces pi () during the whole-class discussion. It is important to save enough time for the discussion about the meaning and value of at the end of this activity.
When students use the equation to complete a table showing diameters and circumferences of various circles, they are making use of the structure of proportional relationships (MP7).
Launch
If time is limited, consider dividing up the calculations, assigning one or two circles to each group of students. If desired, ask students to label the measurements given in the table on the picture of the circles.
Instruct students to use the constant of proportionality that the class estimated in the previous activity. Give students quiet work time followed by whole-class discussion.
Activity
None
Here are five circles. One measurement for each circle is given in the table.
Use the constant of proportionality estimated in the previous activity to complete the table.
diameter (cm)
circumference (cm)
circle A
3
circle B
10
circle C
24
circle D
18
circle E
1
Student Response
Activity Synthesis
The goal of this discussion is to introduce the equation . First, display the table from the activity for all to see. Invite students to share how they calculated the missing values. Record their reasoning on and around the table. Draw students’ attention to their use of multiplication or division by the constant of proportionality.
If no student mentions using an equation of the form , where is the constant of proportionality agreed upon in the previous activity, ask students how we could summarize this relationship with an equation. Display the equation (or whatever value of the class agreed upon) and ask students to describe what this equation means.
Next, ask if students have heard of the number pi or seen the symbol . Tell students that pi is the exact value of the constant of proportionality for this relationship. Explain that the exact value of π is a decimal with infinitely many digits and no repeating pattern, so an approximation is often used. Frequently used approximations for include , 3.14, and 3.14159, but none of these are exactly equal to . Adjust the displayed equation to say .
MLR8 Discussion Supports. To support the transfer of new vocabulary to long-term memory, invite students to chorally repeat these phrases in unison 1–2 times:
“Pi is the constant of proportionality between diameter and circumference.”
“Circumference is pi times diameter.”
“Diameter is circumference divided by pi.”
Advances: Speaking, Listening
Lesson Synthesis
Share with students “Today we measured the diameter and circumference of different circles. We saw that there is a proportional relationship between these quantities. We estimated the constant of proportionality for the relationship.”
To review the relationship between diameter and circumference, consider asking students:
“How did the graph help us decide that there is a proportional relationship between diameter and circumference?” (When we plotted our measurements for diameter and circumference, the points were close to a straight line through .)
“How did the table help us decide that there is a proportional relationship between diameter and circumference?” (When we divided the circumference by the diameter for each row, the quotients were all close to the same value.)
“If we want to write an equation like to represent this relationship, what can we use as , the constant of proportionality?” (The exact constant of proportionality is called pi and written . The value is a little more than 3.)
If desired, use this example to review these concepts:
“A circle has a diameter of 4 cm. What is its circumference?” (approximately 12.4 cm, because )
Student Lesson Summary
There is a proportional relationship between the diameter and circumference of any circle. That means that if we write for circumference and for diameter, we know that , where is the constant of proportionality.
The exact value for the constant of proportionality is called pi, and its symbol is . Some frequently used approximations for are , 3.14, and 3.14159, but none of these is exactly .
A graph of a line in the coordinate plane with the origin labeled O. The horizontal axis is labeled “d” and the numbers 1 through 6 are indicated. The vertical axis is labeled “C” and the numbers 2 through 12, in increments of 2, are indicated. The line begins at the origin, slants upward and to the right, and passes through the point 1 comma pi.
We can use this to estimate the circumference if we know the diameter, and vice versa. For example, using 3.1 as an approximation for , if a circle has a diameter of 4 cm, then the circumference is about , or 12.4 cm.
The relationship between the circumference and the diameter can be written as
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
Arrange students in groups of 2–3. Distribute two circular objects to each group, along with measuring tapes or string and rulers. Consider demonstrating how to measure the circumference, especially if using string and rulers. Wrap a string around the object, make note of where it completes one circle, unwrap the object, and use a ruler to measure the marked length. Encourage students to be as precise as possible as they measure.
Ask students to complete the first two questions in their group, and then to gather additional information from another group (who measured different objects) for the last two questions.
Select students with different strategies, such as those described in the Activity Narrative, to share later.
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Consider giving students one circular object at a time. Check in with students to provide feedback and encouragement after each chunk. Look for students’ methods for measuring circumference. Supports accessibility for: Attention, Social-Emotional Functioning
Activity
None
Your teacher will give you two circular objects.
Measure the diameter and the circumference of each circle to the nearest tenth of a centimeter. Record your measurements in the first two rows of the table.
object
diameter (cm)
circumference (cm)
Plot your diameter and circumference values on the coordinate plane. What do you notice?
A coordinate plane with the origin labeled "O". The horizontal axis is labeled "diameter, in centimeters," and the numbers 0 through 25, in increments of 5, are indicated. The vertical axis is labeled "circumference, in centimeters," and the numbers 0 through 80, in increments of 10, are indicated.
Find out the measurements from another group that measured different objects. Record their values in your table and plot them on your same coordinate plane.
What do you notice about the diameter and circumference values for these four circles?
Student Response
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Building on Student Thinking
Students may try to measure the diameter without going across the widest part of the circle, or may struggle with measuring around the circumference. Mentally check that their measurements divide to get approximately 3 or compare with your own prepared table of data and prompt them to re-measure when their measurements are off by too much. If the circular object has a rim or lip, this could help students keep the measuring tape in place while measuring the circumference.
If students are struggling to see the proportional relationship, remind them of recent examples in which they have seen similar graphs of proportional relationships. Ask them to estimate additional diameter-circumference pairs that would fit the pattern shown in the graph. Based on their graphs, do the values of the circumferences seem to relate to those of the diameters in a particular way? What seems to be that relationship?
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Building on Student Thinking
Some students may multiply the circumference by the constant of proportionality instead of dividing by it. Prompt them to consider whether the diameter can be longer than the circumference of a circle.
Some students may struggle to divide by 3.1 if that is the constant of proportionality decided on in the previous lesson. Ask these students if they could use an easier number as their constant, and allow them to divide by 3 instead. Then ask them how their answer would have changed if they divided by 3.1.
